Bad list assignments for non-$$k$$-choosable $$k$$-chromatic graphs with $$2k+2$$-vertices

It was conjectured by Ohba, and proved by Noel, Reed and Wu that $$k$$-chromatic graphs $$G$$ with $$| V(G)| \le 2k+1$$ are chromatic-choosable. This upper bound on $$| V(G)| $$ is tight: if $$k$$ is even, then $${K}_{3\star (k\unicode{x02215}2+1),1\star (k\unicode{x02215}2-1)}$$ and $${K}_{4,2\star (k-1)}$$ are $$k$$-chromatic graphs with $$2k+2$$ vertices that are not chromatic-choosable. It was proved by Zhu and Zhu that these are the only non-$$k$$-choosable complete $$k$$-partite graphs with $$2k+2$$ vertices. For $$G\in \{{K}_{3\star (k\unicode{x02215}2+1),1\star (k\unicode{x02215}2-1)},{K}_{4,2\star (k-1)}\}$$, a bad list assignment of $$G$$ is a $$k$$-list assignment $$L$$ of $$G$$ such that $$G$$ is not $$L$$-colourable. Bad list assignments for $$G={K}_{4,2\star (k-1)}$$ were characterized by Enomoto, Ohba, Ota and Sakamoto. In this paper, we first give a simpler proof of this result, and then we characterize bad list assignments for $$G={K}_{3\star (k\unicode{x02215}2+1),1\star (k\unicode{x02215}2-1)}$$. Using these results, we characterize all non-$$k$$-choosable $$k$$-chromatic graphs with $$2k+2$$ vertices.